Matching in a family of piecewise affine maps

We consider a class of simple one parameter families of interval maps, and we study how metric (resp. topological) entropy changes as the parameter varies. We show that in many cases the entropy displays a semi-regular behaviour, i.e. it is smooth on an open and dense set. This feature is due to a combinatorial property called matching, which was first observed in the parametric family of α-continued fractions introduced by Nakada and Natsui (2008 Nonlinearity 21 1207–25).